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Sep 18, 2013
09/13

by
Igor Rivin

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In this note we show the n-2-dimensional volumes of codimension 2 faces of an n-dimensional simplex are algebraically independent functions of the lengths of edges. In order to prove this we compute the complete spectrum of a combinatorially interesting graph.

Source: http://arxiv.org/abs/0803.1317v1

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Sep 18, 2013
09/13

by
Igor Rivin

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Motivated by the study of the local extrema of sin(x)/x we define the \emph{Amplitude Modulation} transform of functions defined on (subsets of) the real line. We discuss certain properties of this transform and invert it in some easy cases.

Source: http://arxiv.org/abs/math/0212199v1

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Sep 22, 2013
09/13

by
Igor Rivin

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We begin by studying the surface area of an ellipsoid in n-dimensional Euclidean space as the function of the lengths of the semi-axes. We write down an explicit formula as an integral over the unit sphere in n-dimensions and use this formula to derive convexity properties of the surface area, to give sharp estimates for the surface area of a large-dimensional ellipsoid, to produce asymptotic formulas for the surface area and the \emph{isoperimetric ratio} of an ellipsoid in large dimensions,...

Source: http://arxiv.org/abs/math/0403375v1

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Sep 22, 2013
09/13

by
Igor Rivin

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We obtain sharp bounds for the number of n--cycles in a finite graph as a function of the number of edges, and prove that the complete graph is optimal in more ways than could be imagined. We prove sharp estimates on both the sum of k-th powers of the coordinates and the Lk norm subject to the constraints that the sum of squares of the coordinates is fixed, and that the sum of the coordinates vanishes.

Source: http://arxiv.org/abs/math/0111106v1

On verso of t.p.: Rainbow of verse

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Sep 19, 2013
09/13

by
Igor Rivin

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We give a very short proof of the Golden-Thompson inequality

Source: http://arxiv.org/abs/1010.2193v1

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Sep 21, 2013
09/13

by
Igor Rivin

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We study the moduli space of euclidean structures with cone points on a surface, and describe a decomposition into cells each of which corresponds to a given combinatorial type of Delaunay tessellation. We use some of the ideas to study hyperbolic structures on three-dimensional manifolds

Source: http://arxiv.org/abs/math/9907032v1

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Sep 21, 2013
09/13

by
Igor Rivin

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We show that the number of simple closed geodesics of length bounded by L on a hyperbolic surface of genus g with c cusps and b boundary components grows roughly like L^{6g+2b+2c-6}. This has been conjectured for some time.

Source: http://arxiv.org/abs/math/9907041v2

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Sep 18, 2013
09/13

by
Igor Rivin

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The main result is that every complete finite area hyperbolic metric on a sphere with punctures can be uniquely realized as the induced metric on the surface of a convex ideal polyhedron in hyperbolic 3-space. A number of other observations are included.

Source: http://arxiv.org/abs/math/0005234v1

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Sep 19, 2013
09/13

by
Igor Rivin

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It has been pointed out to the author by David Glickenstein that the proof of the (closely related) Lemmas 1.2 and 3.2 in the title paper is incorrect. The statements of both Lemmas are correct, and the purpose of this note is to give a correct argument. The argument is of some interest in its own right.

Source: http://arxiv.org/abs/math/0302069v2

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0.0

Jun 30, 2018
06/18

by
Igor Rivin

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We study random elements of subgroups (and cosets) of the mapping class group of a closed hyperbolic surface, in part through the properties of their mapping tori. In particular, we study the distribution of the homology of the mapping torus (with rational, integer, and finite field coefficients, the hyperbolic volume (whenever the manifold is hyperbolic), the dilatation of the monodromy, the injectivity radius, and the bottom eigenvalue of the Laplacian on these mapping tori. We also study...

Topics: Dynamical Systems, Mathematics, Geometric Topology

Source: http://arxiv.org/abs/1401.5736

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5.0

Jun 28, 2018
06/18

by
Igor Rivin

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We show that the Galois group of a random monic polynomial %of degree $d>12$ with integer coefficients between $-N$ and $N$ is NOT $S_d$ with probability $\ll \frac{\log^{\Omega(d)}N}{N}.$ Conditionally on NOTbeing the full symmetric group, we have a hierarchy of possibilities each of which has polylog probability of occurring. These results also apply to random polynomials with only a subset of the coefficients allowed to vary. This settles a question going back to 1936.

Topics: Number Theory, Mathematics

Source: http://arxiv.org/abs/1511.06446

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Sep 18, 2013
09/13

by
Igor Rivin

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In his paper "On the Schlafli differential equality", J. Milnor conjectured that the volume of n-dimensional hyperbolic and spherical simplices, as a function of the dihedral angles, extends continuously to the closure of the space of allowable angles. A proof of this has recently been given by F. Luo (see math.GT/0412208). In this paper we give a simple proof of this conjecture, prove much sharper regularity results, and then extend the method to apply to a large class of convex...

Source: http://arxiv.org/abs/math/0502543v2

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1.0

Jun 30, 2018
06/18

by
Igor Rivin

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I describe some deep-seated problems in higher mathematical education, and give some ideas for their solution -- I advocate a move away from the traditional introduction of mathematics through calculus, and towards computation and discrete mathematics.

Topics: Mathematics, History and Overview

Source: http://arxiv.org/abs/1401.0828

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2.0

Jun 30, 2018
06/18

by
Igor Rivin

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We describe extensive computational experiments on spectral properties of random objects - random cubic graphs, random planar triangulations, and Voronoi and Delaunay diagrams of random (uniformly distributed) point sets on the sphere). We look at bulk eigenvalue distribution, eigenvalue spacings, and locality properties of eigenvectors. We also look at the statistics of \emph{nodal domains} of eigenvectors on these graphs. In all cases we discover completely new (at least to this author)...

Topics: High Energy Physics - Theory, Statistical Mechanics, Mathematics, Spectral Theory, Computing...

Source: http://arxiv.org/abs/1410.6771

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1.0

Jun 30, 2018
06/18

by
Igor Rivin

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We give a different perspective on the (by now) classic Basmajian identity, and point out some related results, both in the setting of hyperbolic manifolds, and in the polyhedral setting \emph{without} any group acting. In the new version we give more geometric and combinatorial applications of the main ideas.

Topics: Mathematics, Geometric Topology

Source: http://arxiv.org/abs/1404.1583

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Sep 21, 2013
09/13

by
Igor Rivin

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In this note we show that for any hyperbolic surface S, the number of geodesics of length bounded above by L in the mapping class group orbit of a fixed closed geodesic with a single double point is asymptotic to L raised to the dimension of the Teichmuller space of S. Since closed geodesics with one double point fall into a finite number of orbits under the mapping class group of S, we get the same asympotic estimate for the number of such geodesics of length bounded by L. We also use our...

Source: http://arxiv.org/abs/0901.2543v4

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Sep 18, 2013
09/13

by
Igor Rivin

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We study the surface area of an ellipsoid in n-dimensional Euclidean space as the function of the lengths of their major semi-axes. We write down an explicit formula as an integral over the unit sphere, use the formula to derive convexity properties of the surface area, to give sharp estimates for the surface area of a large-dimensional ellipsoid, to produce asymptotic formulas in large dimensions, and to give an expression for the surface in terms of the Lauricella hypergeometric function.

Source: http://arxiv.org/abs/math/0306387v4

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Sep 18, 2013
09/13

by
Igor Rivin

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We give a survey of some known results and of the many open questions in the study of generic phenomena in geometrically interesting groups.

Source: http://arxiv.org/abs/1211.6509v1

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Jul 20, 2013
07/13

by
Igor Rivin

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We give upper bounds on the numbers of various classes of polynomials reducible over the integers and over integers modulo a prime and on the number of matrices in SL(n), GL(n) and Sp(2n) with reducible characteristic polynomials, and on polynomials with non-generic Galois groups. We use our result to show that a random (in the appropriate sense) element of the mapping class group of a closed surface is pseudo-Anosov, and that a random automorphism of a free group is strongly irreducible (aka...

Source: http://arxiv.org/abs/math/0604489v2

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Sep 18, 2013
09/13

by
Igor Rivin

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We write down estimates for the surface area, and more generally, integral mean curvatures of an ellipsoid E in n-dimensional Euclidean space in terms of the lengths of the major semi-axes. We give applications to estimating the area of parallel surfaces and volume of the tubular neighborhood of E, to the counting of lattice points contained in E and to estimating the shape of the John ellipsoid of a convex body.

Source: http://arxiv.org/abs/math/0306085v2

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Sep 19, 2013
09/13

by
Igor Rivin

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We prove sharp limit theorems on random walks on graphs with values in finite groups. We then apply these results (together with some elementary algebraic geometry, number theory, and representation theory) to finite quotients of lattices in semisimple Lie groups (specifically SL(n,Z) and Sp(2n, Z) to show that a ``random'' element in one of these lattices has irreducible characteristic polynomials (over the integers. The term ``random'' can be defined in at least two ways (in terms of height...

Source: http://arxiv.org/abs/math/0703532v1

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Sep 23, 2013
09/13

by
Igor Rivin

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We study convex sets C of finite (but non-zero volume in Hn and En. We show that the intersection of any such set with the ideal boundary of Hn has Minkowski (and thus Hausdorff) dimension of at most (n-1)/2, and this bound is sharp. In the hyperbolic case we show that for any k

Source: http://arxiv.org/abs/0801.0077v1

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Sep 21, 2013
09/13

by
Igor Rivin

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Given a manifold M, it is natural to ask in how many ways it fibers (we mean fibering in a general way, where the base might be an orbifold -- this could be described as Seifert fibering)There are group-theoretic obstructions to the existence of even one fibering, and in some cases (such as Kahler manifolds or three-dimensional manifolds) the question reduces to a group-theoretic question. In this note we summarize the author's state of knowledge of the subject.

Source: http://arxiv.org/abs/1106.4595v2

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Sep 17, 2013
09/13

by
Igor Rivin

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We write down a one-dimensional integral formula and compute large-n asymptotics for the expectation of the absolute value of the smallest component of a unit vector in n-dimensional Euclidean space. The method is general, and allows to write the mean over the sphere of an homogeneous function in terms of an expectation of a function of independent, identically distributed Gaussians. We also write down an asymptotic formula for the minimum of a large number of identical independent positive...

Source: http://arxiv.org/abs/math/0305252v1

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Sep 19, 2013
09/13

by
Igor Rivin

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We start by studying the distribution of (cyclically reduced) elements of the free groups with respect to their abelianization. We derive an explicit generating function, and a limiting distribution, by means of certain results (of independent interest) on Chebyshev polynomials; we also prove that the reductions $\mod p$ ($p$ -- an arbitrary prime) of these classes are asymptotically equidistributed, and we study the deviation from equidistribution. We extend our techniques to a more general...

Source: http://arxiv.org/abs/math/9911076v2

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Sep 21, 2013
09/13

by
Igor Rivin

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We start by studying the distribution of (cyclically reduced) elements of the free groups Fn with respect to their abelianization (or equivalently, their integer homology class. We derive an explicit generating function, and a limiting distribution, by means of certain results (of independent interest) on Chebyshev polynomials; we also prove that the reductions modulo an arbitrary prime of these classes are asymptotically equidistributed, and we study the deviation from equidistribution. We...

Source: http://arxiv.org/abs/1106.5947v1

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Jul 20, 2013
07/13

by
Igor Rivin

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We obtain sharp bounds for the number of n-cycles in a finite graph as a function of the number of edges, and prove that the complete graph is optimal in more ways than could be imagined. En route, we prove some sharp estimates on power sums.

Source: http://arxiv.org/abs/math/9910093v1

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Sep 20, 2013
09/13

by
Igor Rivin

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We define a class of multivariate Laurent polynomials closely related to Chebyshev polynomials, and prove the simple but somewhat surprising (in view of the fact that the signs of the coefficients of the Chebyshev polynomials themselves alternate) result that their coefficients are non-negative. We further show that a Central Limit Theorem holds for our polynomials.

Source: http://arxiv.org/abs/math/0301241v1

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Sep 18, 2013
09/13

by
Igor Rivin

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An extra large metric is a spherical cone metric with all cone angles greater than 2 pi and every closed geodesic longer than 2pi. We show that every two-dimensional extra large metric can be triangulated with vertices at cone points only. The argument implies the same result for Euclidean and hyperbolic cone metrics, and can be modified to show a similar result for higher-dimensional extra-large metrics. The extra-large hypothesis is necessary.

Source: http://arxiv.org/abs/math/0509320v1

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2.0

Jun 29, 2018
06/18

by
Igor Rivin

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We study random knots, which we define as a triple of random periodic functions (where a random function is a random trigonometric series, \[f(\theta) = \sum_{k=1}^\infty a_k \cos (k \theta) +b_k (\sin k \theta),\] with $a_k, b_k$ are independent gaussian random variables with mean $0$ and variance $\sigma(k)^2$ - our results will depend on the functional dependence of $\sigma$ on $k.$ In particular, we show that if $\sigma(k) = k^\alpha,$ with $\alpha < -3/2,$ then the probability of...

Topics: Probability, Geometric Topology, Statistical Mechanics, Condensed Matter, Mathematics

Source: http://arxiv.org/abs/1607.05239

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Sep 19, 2013
09/13

by
Igor Rivin

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In 1957, Chandler Davis proved that unitarily invariant convex functions on the space of hermitian matrices are precisely those which are convex and symmetrically invariant on the set of diagonal matrices. We give a simple perturbation theoretic proof of this result. (Davis' argument was also very short, though based on completely different ideas).

Source: http://arxiv.org/abs/math/0208223v3

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Sep 20, 2013
09/13

by
Igor Rivin

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We define a class of multivariate Laurent polynomials closely related to Chebyshev polynomials, and prove the simple but somewhat surprising (in view of the fact that the signs of the coefficients of the Chebyshev polynomials themselves alternate) result that their coefficients are non-negative.

Source: http://arxiv.org/abs/math/0301210v1

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Sep 22, 2013
09/13

by
Igor Rivin

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Motivated by a probabilistic analysis of a simple game (itself inspired by a problem in computational learning theory) we introduce the \emph{moment zeta function} of a probability distribution, and study in depth some asymptotic properties of the moment zeta function of those distributions supported in the interval [0, 1]. One example of such zeta functions is Riemann's zeta function (which is the moment zeta function of the uniform distribution in [0, 1]. For Riemann's zeta function we are...

Source: http://arxiv.org/abs/math/0201109v1

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Jun 28, 2018
06/18

by
Igor Rivin

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We describe some studies related to the frequency of prime values of integer polynomials.

Topics: Number Theory, Mathematics

Source: http://arxiv.org/abs/1508.07821

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Jul 20, 2013
07/13

by
Igor Rivin

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We study the convergence speed of the batch learning algorithm, and compare its speed to that of the memoryless learning algorithm and of learning with memory (as analyzed in joint work with N. Komarova). We obtain precise results and show in particular that the batch learning algorithm is never worse than the memoryless learning algorithm (at least asymptotically). Its performance vis-a-vis learning with full memory is less clearcut, and depends on certainprobabilistic assumptions. These...

Source: http://arxiv.org/abs/cs/0107033v1

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Sep 22, 2013
09/13

by
Igor Rivin

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We analyze completely the convergence speed of the \emph{batch learning algorithm}, and compare its speed to that of the memoryless learning algorithm and of learning with memory. We show that the batch learning algorithm is never worse than the memoryless learning algorithm (at least asymptotically). Its performance \emph{vis-a-vis} learning with full memory is less clearcut, and depends on certain probabilistic assumptions.

Source: http://arxiv.org/abs/cs/0201009v1

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Sep 23, 2013
09/13

by
Igor Rivin

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Let A be an n by n matrix with determinant 1. We show that for all n > 2 there exist dimensional strictly positive constants C_n such that the average over the orthogonal group of log rho(A X) d X > C_n log ||A||, where ||A|| denotes the operator norm of A (which equals the largest singular value of A), rho denotes the spectral radius, and the integral is with respect to the Haar measure on O_n The same result (with essentially the same proof) holds for the unitary group U_n in place of...

Source: http://arxiv.org/abs/math/0312048v2

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Sep 24, 2013
09/13

by
Igor Rivin

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For a linear transformation A from Rn to Rn, we give sharp bounds for the average distortion of A, that is, the average value of log of the euclidean norm of Au over all unit vectors u. This is closely related to the results of the author's paper math.DS/0312048.

Source: http://arxiv.org/abs/math/0412260v1

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Sep 19, 2013
09/13

by
Igor Rivin

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We observe that a sharp result on the exponential growth rate of the number of primitive elements exists for the free group on two generators.

Source: http://arxiv.org/abs/math/0302083v1

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Sep 19, 2013
09/13

by
Igor Rivin

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We consider the following situation: G is a finite directed graph, where to each vertex of G is assigned an element of a finite group Gamma. We consider all walks of length N on G, starting from v_i and ending at v_j To each such walk $w$ we assign the element of Gamma equal to the product of the elements along the walk. The set of all walks of length N from v_i to v_j thus induces a probability distribution $F_N on Gamma In previous work we have given necessary and sufficient conditions for...

Source: http://arxiv.org/abs/math/0703533v1

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Sep 22, 2013
09/13

by
Igor Rivin

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We investigate the space of simplices in Euclidean Space

Source: http://arxiv.org/abs/math/0308239v1

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Sep 23, 2013
09/13

by
Igor Rivin

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If a (cusped) surface S admits an ideal triangulation T with no shears, we show an efficient algorithm to give S as a quotient of hypebolic plane by a subgroup of PSL(2, Z). The algorithm runs in time O(n log n), where n is the number of triangles in the triangulation T. The algorithm generalizes to producing fundamental groups of general surfaces and geometric manifolds of higher dimension.

Source: http://arxiv.org/abs/math/0510613v1

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Sep 19, 2013
09/13

by
Henry Cejtin; Igor Rivin

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We describe an efficient algorithm to write any element of the alternating group A_n as a product of two n-cycles (in particular, we show that any element of A_n can be so written -- a result of E. A. Bertram). An easy corollary is that every element of A_n is a commutator in the symmetric group S_n.

Source: http://arxiv.org/abs/math/0303036v2

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Sep 18, 2013
09/13

by
Greg McShane; Igor Rivin

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We define a norm on homology of punctured tori equipped with a complete hyperbolic metric of finite volume and use it to find asymptotics on the growth of the number of simple geodesics of bounded length.

Source: http://arxiv.org/abs/math/0005222v1

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Sep 21, 2013
09/13

by
Ilya Kapovich; Igor Rivin

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A remarkable result of McShane states that for a punctured torus with a complete finite volume hyperbolic metric we have \[ \sum_{\gamma} \frac{1}{e^{\ell(\gamma)}+1}={1/2} \] where $\gamma$ varies over the homotopy classes of essential simple closed curves and $\ell(\gamma)$ is the length of the geodesic representative of $\gamma$. We prove that there is no reasonable analogue of McShane's identity for the Culler-Vogtmann outer space of a free group.

Source: http://arxiv.org/abs/math/0602291v1

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Sep 23, 2013
09/13

by
Robin Pemantle; Igor Rivin

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In this note we initiate the probabilistic study of the critical points of polynomials of large degree with a given distribution of roots. Namely, let f be a polynomial of degree n whose zeros are chosen IID from a probability measure mu on the complex numbers. We conjecture that the zero set of f' always converges in distribution to mu as n goes to infinity. We prove this for measures with finite one-dimensional energy. When mu is uniform on the unit circle this condition fails. In this...

Source: http://arxiv.org/abs/1109.5975v1

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Sep 22, 2013
09/13

by
Dmitry Jakobson; Igor Rivin

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We define a number of natural (from geometric and combinatorial points of view) deformation spaces of valuations on finite graphs, and study functions over these deformation spaces. These functions include both direct metric invariants (girth, diameter), and spectral invariants (the determinant of the Laplace operator, or complexity; bottom non-zero eigenvalue of the Laplace operator). We show that almost all of these functions are, surprisingly, convex, and we characterize the valuations...

Source: http://arxiv.org/abs/math/0001169v1

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Sep 18, 2013
09/13

by
Natalia Komarova; Igor Rivin

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We study the convergence properties of a pair of learning algorithms (learning with and without memory). This leads us to study the dominant eigenvalue of a class of random matrices. This turns out to be related to the roots of the derivative of random polynomials (generated by picking their roots uniformly at random in the interval [0, 1], although our results extend to other distributions). This, in turn, requires the study of the statistical behavior of the harmonic mean of random variables...

Source: http://arxiv.org/abs/math/0105235v3